Kinetic Theory of Jet Dynamics in the Stochastic Barotropic and 2D Navier-Stokes Equations
Tóm tắt
We discuss the dynamics of zonal (or unidirectional) jets for barotropic flows forced by Gaussian stochastic fields with white in time correlation functions. This problem contains the stochastic dynamics of 2D Navier-Stokes equation as a special case. We consider the limit of weak forces and dissipation, when there is a time scale separation between the inertial time scale (fast) and the spin-up or spin-down time (large) needed to reach an average energy balance. In this limit, we show that an adiabatic reduction (or stochastic averaging) of the dynamics can be performed. We then obtain a kinetic equation that describes the slow evolution of zonal jets over a very long time scale, where the effect of non-zonal turbulence has been integrated out. The main theoretical difficulty, achieved in this work, is to analyze the stationary distribution of a Lyapunov equation that describes quasi-Gaussian fluctuations around each zonal jet, in the inertial limit. This is necessary to prove that there is no ultraviolet divergence at leading order, in such a way that the asymptotic expansion is self-consistent. We obtain at leading order a Fokker–Planck equation, associated to a stochastic kinetic equation, that describes the slow jet dynamics. Its deterministic part is related to well known phenomenological theories (for instance Stochastic Structural Stability Theory) and to quasi-linear approximations, whereas the stochastic part allows to go beyond the computation of the most probable zonal jet. We argue that the effect of the stochastic part may be of huge importance when, as for instance in the proximity of phase transitions, more than one attractor of the dynamics is present.
Tài liệu tham khảo
Bakas, N., Ioannou, P.: A theory for the emergence of coherent structures in beta-plane turbulence (2013). arXiv:1303.6435
Berhanu, M., Monchaux, R., Fauve, S., Mordant, N., Pétrélis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Marié, L., Ravelet, F., Bourgoin, M., Odier, P., Pinton, J.-F., Volk, R.: Magnetic field reversals in an experimental turbulent dynamo. Europhys. Lett. 77(5), 59001 (2007)
Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton (1987). 747 p.
Boffetta, G., Ecke, R.E.: Two-dimensional turbulence. Annu. Rev. Fluid Mech. 44, 427–451 (2012)
Bouchet, F.: Stochastic process of equilibrium fluctuations of a system with long-range interactions. Phys. Rev. E 70(3), 036113 (2004)
Bouchet, F., Dauxois, T.: Prediction of anomalous diffusion and algebraic relaxations for long-range interacting systems, using classical statistical mechanics. Phys. Rev. E 72(4), 045103 (2005)
Bouchet, F., Morita, H.: Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations. Phys D, Nonlinear Phenom. 239, 948–966 (2010)
Bouchet, F., Simonnet, E.: Random changes of flow topology in two-dimensional and geophysical turbulence. Phys. Rev. Lett. 102(9), 094504 (2009)
Bouchet, F., Venaille, A.: Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295 (2012)
Bouchet, F., Dauxois, T.: Kinetics of anomalous transport and algebraic correlations in a long-range interacting system. J. Phys. Conf. Ser. 7, 34 (2005)
Bréhier, C.-E.: Strong and weak order in averaging for SPDEs. Stoch. Proc. Appl. (2012)
Bricmont, J., Kupiainen, A., Lefevere, R.: Ergodicity of the 2D Navier-Stokes equations with random forcing. Commun. Math. Phys. 224, 65–81 (2001)
Campa, A., Dauxois, T., Ruffo, S.: Statistical mechanics and dynamics of solvable models with long-range interactions. Phys. Rep. 480, 57–159 (2009)
Case, K.M.: Stability of inviscid plane Couette flow. Phys. Fluids 3, 143–148 (1960)
Chavanis, P.H.: Quasilinear theory of the 2D Euler equation. Phys. Rev. Lett. 84, 5512–5515 (2000)
Chavanis, P.H.: Kinetic theory of point vortices: diffusion coefficient and systematic drift. Phys. Rev. E 64(2), 026309 (2001)
Chavanis, P.H.: Statistical mechanics of two-dimensional vortices and stellar systems. In: Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds.) Dynamics and Thermodynamics of Systems with Long Range Interactions. Lecture Notes in Physics, vol. 602, pp. 208–289. Springer, Berlin (2002)
Chertkov, M., Connaughton, C., Kolokolov, I., Lebedev, V.: Dynamics of energy condensation in two-dimensional turbulence. Phys. Rev. Lett. 99, 084501 (2007)
Constantinou, N.C., Ioannou, P.J., Farrell, B.F.: Emergence and equilibration of jets in beta-plane turbulence (2012). arXiv:1208.5665
Da Prato, G., Debussche, A.: Ergodicity for the 3d stochastic Navier–Stokes equations. J. Math. Pures Appl. 82(8), 877–947 (2003)
Danilov, S., Gurarie, D.: Scaling, spectra and zonal jets in beta-plane turbulence. Phys. Fluids 16, 2592 (2004)
DelSole, T., Farrell, B.F.: The quasi-linear equilibration of a thermally maintained, stochastically excited jet in a quasigeostrophic model. J. Atmos. Sci. 53(13), 1781–1797 (1996)
Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, 2nd edn. Cambridge University Press, Cambridge (2004)
Dritschel, D.G., McIntyre, M.E.: Multiple jets as PV staircases: the Phillips effect and the resilience of eddy-transport barriers. J. Atmos. Sci. 65, 855 (2008)
Dubin, D.H.E., O’Neil, T.M.: Two-dimensional guiding-center transport of a pure electron plasma. Phys. Rev. Lett. 60(13), 1286–1289 (1988)
Farrell, B.F., Ioannou, P.J.: Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64, 3652 (2007)
Farrell, B.F., Ioannou, P.J.: Structural stability of turbulent jets. J. Atmos. Sci. 60, 2101–2118 (2003)
Ferrario, B.: Ergodic results for stochastic Navier-Stokes equation. Stoch. Int. J. Probab. Stoch. Process. 60(3–4), 271–288 (1997)
Flandoli, F., Maslowski, B.: Ergodicity of the 2-d Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172(1), 119–141 (1995)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer, New York (1984)
Galperin, B., Sukoriansky, S., Dikovskaya, N.: Geophysical flows with anisotropic turbulence and dispersive waves: flows with a β-effect. Ocean Dyn. 60(2), 427–441 (2010)
Galperin, B., Sukoriansky, S., Huang, H.-P.: Universal n spectrum of zonal flows on giant planets. Phys. Fluids 13, 1545 (2001)
Gardiner, C.W.: Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics. Springer, Berlin (1994). 2nd ed. 1985. Corr. 3rd printing 1994
Gourcy, M.: A large deviation principle for 2d stochastic Navier–Stokes equation. Stoch. Process. Appl. 117(7), 904–927 (2007)
Hairer, M., Mattingly, J.C.: Ergodicity of the 2d Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 993–1032 (2006)
Hairer, M., Mattingly, J.C.: Spectral gaps in Wasserstein distances and the 2d stochastic Navier–Stokes equations. Ann. Probab. 36(6), 2050–2091 (2008)
Jaksic, V., Nersesyan, V., Pillet, C.-A., Shirikyan, A.: Large deviations from a stationary measure for a class of dissipative PDE’s with random kicks (2012). arXiv:1212.0527
Kasahara, A.: Effect of zonal flows on the free oscillations of a barotropic atmosphere. J. Atmos. Sci. 37, 917–929 (1980)
Khasminskii, R.Z.: On an averaging principle for Ito stochastic differential equations. Kybernetika 4, 260–279 (1968)
Kuksin, S.B.: The Eulerian limit for 2D statistical hydrodynamics. J. Stat. Phys. 115, 469–492 (2004)
Kuksin, S., Penrose, O.: A family of balance relations for the two-dimensional Navier-Stokes equations with random forcing. J. Stat. Phys. 118(3–4), 437–449 (2005)
Kuksin, S., Shirikyan, A.: Ergodicity for the randomly forced 2d Navier–Stokes equations. Math. Phys. Anal. Geom. 4(2), 147–195 (2001)
Kuksin, S.B., Piatnitski, A.L.: Khasminskii–Whitham averaging for randomly perturbed KdV equation. J. Math. Pures Appl. 89(4), 400–428 (2008)
Kuksin, S.B., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cambridge (2012)
Landau, L.D., Lifshitz, E.M.: Statistical Physics. Course of Theoretical Physics, vol. 5. Pergamon Press, New York (1980)
Loxley, P.N., Nadiga, B.T.: Bistability and hysteresis of maximum-entropy states in decaying two-dimensional turbulence. Phys. Fluids 25, 015113 (2013)
Maassen, S.R., Clercx, H.J.H., Van Heijst, G.J.F.: Self-organization of decaying quasi-two-dimensional turbulence in stratified fluid in rectangular containers. J. Fluid Mech. 495, 19–33 (2003)
Majda, A.J., Wang, X.: Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press, Cambridge (2006)
Majda, A.J., Wang, X.: The emergence of large-scale coherent structure under small-scale random bombardments. Commun. Pure Appl. Math. 59(4), 467–500 (2006)
Marston, J.B.: Statistics of the general circulation from cumulant expansions. Chaos 20(4), 041107 (2010)
Marston, B.: Looking for new problems to solve? Consider the climate. Phys. Online J. 4, 20 (2011)
Marston, J.B., Conover, E., Schneider, T.: Statistics of an unstable barotropic jet from a cumulant expansion. J. Atmos. Sci. 65, 1955 (2008)
Mattingly, J.C., Sinai, Y.G.: An elementary proof of the existence and uniqueness theorem for the Navier–Stokes equations. Commun. Contemp. Math. 1(04), 497–516 (1999)
Miller, J.: Statistical mechanics of Euler equations in two dimensions. Phys. Rev. Lett. 65(17), 2137–2140 (1990)
Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207, 29–201 (2011)
Nardini, C., Gupta, S., Ruffo, S., Dauxois, T., Bouchet, F.: Kinetic theory for non-equilibrium stationary states in long-range interacting systems. J. Stat. Mech. Theory Exp. 1, L01002 (2012)
Nardini, C., Gupta, S., Ruffo, S., Dauxois, T., Bouchet, F.: Kinetic theory of nonequilibrium stochastic long-range systems: phase transition and bistability. J. Stat. Mech. Theory Exp. 2012(12), P12010 (2012)
Nazarenko, S.: On exact solutions for near-wall turbulence theory. Phys. Lett. A 264, 444–448 (2000)
Nazarenko, S., Kevlahan, N.K.-R., Dubrulle, B.: WKB theory for rapid distortion of inhomogeneous turbulence. J. Fluid Mech. 390(1), 325–348 (1999)
Nazarenko, S., Kevlahan, N.K.-R., Dubrulle, B.: Nonlinear RDT theory of near-wall turbulence. Phys D, Nonlinear Phenom. 139(1), 158–176 (2000)
Nicholson, D.: Introduction to Plasma Theory. Wiley, New York (1983)
O’Gorman, P.A., Schneider, T.: Recovery of atmospheric flow statistics in a general circulation model without nonlinear eddy-eddy interactions. Geophys. Res. Lett. 34(22) (2007)
Parker, J.B., Krommes, J.A.: Zonal flow as pattern formation: Merging jets and the ultimate jet length scale (2013). arXiv:1301.5059
Pedlosky, J.: Geophysical Fluid Dynamics (1982)
Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)
Ravelet, F., Marié, L., Chiffaudel, A., Daviaud, F.: Multistability and memory effect in a highly turbulent flow: experimental evidence for a global bifurcation. Phys. Rev. Lett. 93(16), 164501 (2004)
Robert, R.: Etats d’équilibre statistique pour l’écoulement bidimensionnel d’un fluide parfait. C. R. Acad. Sci., Ser. 1 Math. 311, 575–578 (1990)
Robert, R.: A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys. 65, 531–553 (1991)
Schmeits, M.J., Dijkstra, H.A.: Bimodal behavior of the Kuroshio and the Gulf stream. J. Phys. Oceanogr. 31, 3435–3456 (2001)
Shirikyan, A.: Exponential mixing for 2d Navier-Stokes equations perturbed by an unbounded noise. J. Math. Fluid Mech. 6(2), 169–193 (2004)
Sommeria, J.: Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168 (1986)
Srinivasan, K., Young, W.R.: Zonostrophic instability. J. Atmos. Sci. 69(5), 1633–1656 (2011)
Sritharana, S.S., Sundarb, P.: Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise. Stoch. Process. Appl. 116, 1636–1659 (2006)
Tobias, S.M., Marston, J.B.: Direct statistical simulation of out-of-equilibrium jets. Phys. Rev. Lett. 110(10), 104502 (2013)
Vallis, G.K.: Atmospheric and Oceanic Fluid Dynamics (2006)
Weeks, E.R., Tian, Y., Urbach, J.S., Ide, K., Swinney, H.L., Ghil, M.: Transitions between blocked and zonal flows in a rotating annulus. Science 278, 1598 (1997)
Weinan, E., Mattingly, J.C.: Ergodicity for the Navier-Stokes equation with degenerate random forcing: finite-dimensional approximation. Commun. Pure Appl. Math. 54, 1386–1402 (2001)
Yamaguchi, Y.Y., Bouchet, F., Dauxois, T.: Algebraic correlation functions and anomalous diffusion in the Hamiltonian mean field model. J. Stat. Mech. 1, 20 (2007)
Yin, Z., Montgomery, D.C., Clercx, H.J.H.: Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of “patches” and “points”. Phys. Fluids 15, 1937–1953 (2003)